Development and Astrophysical Applications of a Parallel Smoothed Particle Hydrodynamics Code with MPI

Smoothed Particle Hydrodynamics SPH is a particle method to sim ulate compressible uids First we present a brief introduction to SPH where we focus on an approach to treat the physical viscosity Then we describe in detail the basic principles of our parallel implementation of the SPH method The e ciency of the code on Cray T E and IBM SP is discussed In the last part we present a short introduction to accretion disks in interacting binary stars and report on some new results in that eld achieved with our code Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics SPH is a grid free Lagrangian method for solving the hydrodynamic equations numerically It was introduced in by Gingold Monaghan and by Lucy The method is especially suited for compressible ows involving free boundaries a commonplace sit uation in astrophysics In SPH the matter distribution is divided into small overlapping mass packets so called particles which do not exchange matter The particles move in space following the motion of the uid while interacting with their neighbor particles Basic Principles The system of hydrodynamic equations i e the Navier Stokes equation the equation of continuity and the energy equation together with an equation of state form a system of coupled partial di erential equations Applying the SPH formalism these are transformed into a set of ordinary di erential equa tions ODEs in two steps These ODEs are subsequently integrated numer ically using standard methods First the eld quantities f r are smoothed i e they are replaced by the convolution f r Z f r W jr r j h dV Stefan Kunze Erik Schnetter and Roland Speith with a di erentiable kernel W which is normalized to unity The smoothing length h is a measure of the compact support of the kernel function leading to f r Z f r W jr r j h dV O h A common choice for the kernel is a cubic spline see e g Monaghan In the second step the above integrals are evaluated at the particle posi tions ri and are approximated by sums fi f ri X j Vjf rj W jri rj j h The quantity Vj describes the volume that is represented by the particle j Spatial derivatives of eld quantities at the particle positions can be approx imated by rif ri X j Vjf rj riW jri rj j h Typical particle quantities are the position ri the velocity vi the particle mass mi Vi i where i is the mass density the internal energy ei etc SPH Representation of the Hydrodynamic Equations The particles follow the motion of the uid dri dt vi and do not exchange matter dmi dt Therefore the continuity equation is automatically satis ed With the abbreviationWij W jri rj j h the particle density i can be calculated from i X